Sequential closedness of Boolean algebras of projections in Banach spaces

Tom 167 / 2005

D. H. Fremlin, B. de Pagter, W. J. Ricker Studia Mathematica 167 (2005), 45-62 MSC: Primary 46E27, 47B40; Secondary 46B26, 46B42. DOI: 10.4064/sm167-1-4


Complete and $\sigma $-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for $\sigma $-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a $\sigma $-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).


  • D. H. FremlinDepartment of Mathematics
    University of Essex
    Wivenhoe Park
    Colchester CO4 3SQ, United Kingdom
  • B. de PagterDepartment of Applied Mathematical Analysis
    Faculty EEMCS
    Delft University of Technology
    Mekelweg 4
    2628CD Delft, The Netherlands
  • W. J. RickerMathematisch-Geographische Fakultät
    Katholische Universität Eichstätt-Ingolstadt
    D-85071 Eichstätt, Germany

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